> I can't understand why we need to use a ground plane at 1/4 wavelength
> and 1/2 wavelength not needed.

A full dipole has voltage and current waves like this: -

[![enter image description here][1]][1]

Picture from [Wikipedia](https://en.wikipedia.org/wiki/Dipole_antenna).

Now, if you focus your eye at dead centre of the picture you will see that the voltage is always zero volts. This is because a dipole is optimally driven with a balanced voltage source (\$V_O\$). A balanced voltage source is preferred for a dipole antenna. In fact, the voltage and electric field is zero all along the length of the green line below: -

[![enter image description here][2]][2]

This means you can optionally regard that green line as earth (providing the antenna is driven in a balanced way).

Now if you were to cut the above picture in half you'd have a quarter wave monopole driven with an unbalanced voltage source. An unbalanced voltage source is one that has typically 0 volts on one leg while the other leg does the voltage driving: -

[![enter image description here][3]][3]

And, not surprisingly, it has one half of the impedance presented by the half wave dipole. But, to keep the same radiation pattern you need to "force" an earth plane that does what the green line does.

  [1]: https://i.sstatic.net/L20T3.gif
  [2]: https://i.sstatic.net/CgmEJ.gif
  [3]: https://i.sstatic.net/1xbyT.jpg